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# Crouching AGM, hidden modularity

Crouching AGM, hidden modularity
Shaun Cooper, Jesús Guillera, Armin Straub, Wadim Zudilin — Preprint — 2016

## Abstract

Special arithmetic series $$f(x)=\sum_{n=0}^\infty c_nx^n$$, whose coefficients $$c_n$$ are normally given as certain binomial sums, satisfy self-replicating' functional identities. For example, the equation $$\frac1{(1+4z)^2}\,f\biggl(\frac z{(1+4z)^3}\biggr) =\frac1{(1+2z)^2}\,f\biggl(\frac{z^2}{(1+2z)^3}\biggr)$$ generates a modular form $$f(x)$$ of weight 2 and level 7, when a related modular parametrization $$x=x(\tau)$$ is properly chosen. In this note we investigate the potential of describing modular forms by such self-replicating equations as well as applications of the equations that do not make use of the modularity. In particular, we outline a new recipe of generating AGM-type algorithms for computing $$\pi$$ and other related constants. Finally, we indicate some possibilities to extend the functional equations to a two-variable setting.

@article{crouching-agm-2016,
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